Problem 37 - Beginning and Intermediate Algebra

Question

Solve using the square root property. Simplify all radicals. $$ x^{2}-12=0 $$

Step-by-Step Solution

Answer

x = \pm 2\sqrt{3}

Step 1: Isolate the variable term

Add 12 to both sides of the equation to isolate the quadratic term: \[ x^2 - 12 + 12 = 12 \]This simplifies to: \[ x^2 = 12 \]

Step 2: Apply the square root property

Take the square root of both sides to solve for \(x\). Remember to consider both the positive and negative roots:\[ \sqrt{x^2} = \pm \sqrt{12} \]This simplifies to:\[ x = \pm \sqrt{12} \]

Step 3: Simplify the radical

Express \( \sqrt{12} \) in its simplest form by factoring out the perfect square:\[ \sqrt{12} = \sqrt{4 \times 3} = \sqrt{4} \cdot \sqrt{3} = 2\sqrt{3} \]Thus, the solutions are: \[ x = \pm 2\sqrt{3} \]

Key Concepts

square root property

The square root property is a useful tool in solving quadratic equations. Essentially, it states that if you have an equation of the form \( x^2 = k \), you can solve for \( x \) by taking the square root of both sides. This results in: \( x = \pm \sqrt{k} \). In mathematical notation, \( \pm \) means 'plus or minus'. This signifies that there are always two solutions: a positive root and a negative root.

For example, in the equation \( x^2 = 12 \), applying the square root property gives: \( x = \pm \sqrt{12} \). Remember, you must always consider both positive and negative possibilities. This ensures you capture all potential solutions to the equation.

Applying the square root property is straightforward:

• Isolate the squared term (move other terms to the opposite side).
• Take the square root of both sides of the equation.
• Include both positive and negative roots in your solutions.

Once you understand this property, it becomes a powerful method to solve various quadratic equations.

simplifying radicals

Simplifying radicals involves breaking down a radical expression into its simplest form. This makes the expression easier to understand and use in further calculations. To simplify a radical like \( \sqrt{12} \):

• First, factor the number under the radical into its prime factors. For 12, you get: \(12 = 4 \times 3 \).
• Next, find the perfect squares within those factors. Here, \( 4 \) is a perfect square since \( \sqrt{4} = 2 \).
• Separate the perfect square from the other factors. So, \( \sqrt{12} = \sqrt{4 \times 3} \).
• Simplify by taking the square root of the perfect square part. This gives: \( \sqrt{4} \times \sqrt{3} = 2 \sqrt{3} \).

By following these steps, you can simplify any radical expression. Remember to always look for the largest perfect square factor within the radicand (the number under the square root sign).

This method ensures your solutions are in their simplest and most understandable form.

factoring perfect squares

Factoring perfect squares means rewriting a squared term as a product of its square root. This technique is often used to simplify equations and expressions.

Consider a quadratic like \( x^2 - 12 \). Although 12 is not a perfect square, let's say you had \( x^2 - 16 \). You could rewrite 16 as \( 4^2 \), a perfect square. This simplifies to: \( x^2 - 4^2 \). Now, apply the difference of squares formula: \( x^2 - a^2 = (x - a)(x + a) \).
In this case, \( x^2 - 4^2 = (x - 4)(x + 4) \).

Perfect square trinomials also use this concept. For example, \( x^2 + 6x + 9 \) can be factored as: \( (x + 3)^2 \), because: - The first term is a square: \( x^2 \). - The last term is a square: \( 9 = 3^2 \). - The middle term is twice the product of the first and last terms’ square roots: \( 2 \times x \times 3 = 6x \). The better you get at recognizing perfect squares, the quicker you can factor and solve quadratic problems.